practice_final_solns

practice_final_solns - PSTAT 120A, Spring 2010, Practice...

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PSTAT 120A, Spring 2010, Practice Final. (1) (a) Let X and Y be independent random variables each having the probability mass function p ( x ) = 1 2 ± 2 3 ² x , x = 1 , 2 , 3 ,... Find P ( X = 1 ,Y = 3) and P ( X + Y = 3). (b) Let X and Y be two independent random variables with the same probability density function given by f ( x ) = ( e - x if 0 < x < 0 elsewhere. Show that g , the probability density function of X/Y , is given by g ( t ) = ( 1 (1+ t ) 2 if 0 < t < 0 t 0 . Solution: (a) Since X and Y are independent with same p.m.f., P ( X = 1 ,Y = 3) = P ( X = 1) · P ( Y = 3) = p (1) p (3) = 1 4 ± 2 3 ² 4 = 4 / 81 . P ( X + Y = 3) = P ( X = 1 ,Y = 2) + P ( X = 2 ,Y = 1) = 2 · 1 4 ± 2 3 ² 3 = 4 / 27 . (b) Since X and Y are independent with the same pdf, the joint density function for X and Y is f X,Y ( x,y ) = f ( x ) · f ( y ) i.e. f X,Y ( x,y ) = ( e - ( x + y ) if 0 < x < , 0 < y < 0 elsewhere. We first find the distribution function for Z = X/Y . For z > 0, F Z ( z ) = P ( Z z ) = P ( X/Y z ) = ZZ { ( x,y ): x zy } f X,Y ( x,y ) dxdy = Z 0 Z zy 0 e - x e - y dxdy = Z 0 ( e - y - e - ( z +1) y ) dy = - e - y + e - ( z +1) y z + 1 ³ ³ ³ ³ 0 = 1 - 1 z + 1 . f X/Y ( z ) = d dz F Z ( z ) = 1 ( z + 1) 2 for z > 0 and f X/Y ( z ) = 0 for z 0. 1
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2 (2) According to the US National Center for Health Statistics, 25.2% of males and 23.6% of females never eat breakfast. Suppose that random samples of 200 men and 200 women are chosen. Approximate the probability that (a) at least 110 of these 400 people never eat breakfast; (b) the number of the women who never eat breakfast is at least as large as the number of men who never eat breakfast. Solution: Let X and Y denote, respectively, the number of males and females in the sample that never eat breakfast. Since E [ X ] = 50 . 4 ,V ar ( X ) = 37 . 6992 ,E [ Y ] = 47 . 2 ,V ar ( Y ) = 36 . 0608 it follows from the normal approximation to the binomial that X is approxi-
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This note was uploaded on 06/05/2010 for the course STAT PStat 120a taught by Professor Rohinikumar during the Spring '10 term at UCSB.

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practice_final_solns - PSTAT 120A, Spring 2010, Practice...

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